Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti)de Sitter groups

The κ-deformation of the 2+1 anti-de Sitter, Poincaré and de Sitter groups is studied within a unified approach by considering explicitly the curvature of the spacetime (cosmological constant) as a parameter. The Drinfel’d-double construction as well as the Poisson–Lie structure underlying the κ-deformation are explicitly given, and the three quantum groups are obtained as Weyl quantizations of such Poisson–Lie algebras. As a consequence, the non-commutative 2+1 spacetimes and 4D spaces of worldlines are derived. The former generalize the κ-Minkowski space to the (anti)de Sitter ones, and the latter can be interpreted as a new possibility to introduce noncommutative velocity spaces. Furthermore, provided that the deformation parameter is identified with the Planck length, quantum (anti)de Sitter algebras are presented both in the known basis arising in 2+1 quantum gravity and in a new one which generalizes the bicrossproduct κ-Poincaré basis. Finally, the existence of a kind of ‘duality’ between the cosmological constant and the Planck length is also commented.